7.14.28 problem 28

Internal problem ID [453]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.2 (Series solution near ordinary points). Problems at page 216
Problem number : 28
Date solved : Tuesday, March 04, 2025 at 11:24:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+{\mathrm e}^{-x} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 49

Order:=6; 
ode:=diff(diff(y(x),x),x)+exp(-x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{40} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 56

ode=D[y[x],{x,2}]+Exp[-x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {x^5}{60}+\frac {x^4}{12}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {x^5}{40}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.741 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*exp(-x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4} e^{- 2 x}}{24} - \frac {x^{2} e^{- x}}{2} + 1\right ) + C_{1} x \left (- \frac {x^{2} e^{- x}}{6} + 1\right ) + O\left (x^{6}\right ) \]